Sufficient documentation for educational purpose license for USA-Cuba flights?

Cuban side has been addressed by both great answers above, which I concur with. Which leaves us with the US side only.

First, travel ban to Cuba (in reality it is ban to spend money in Cuba without license, as the government cannot prohibit travel per se) is only affecting persons subject to U.S. jurisdiction. What is jurisdiction? This means you have to be US citizen (permanent residents also are sometime considered, although this is questionable), or be physically present where US government exercises control. Otherwise you’re not under US jurisdiction, and thus the law does not apply to you. Notably having a US visa doesn’t by itself subject you to US jurisdiction outside the US, and thus there could be no possible restrictions if you took a connecting flight such as US-Mexico-Cuba.

However in your case you’re traveling on a direct flight, and under affidavit. Generally an affidavit alone is enough (if not, you’d be asked to submit supporting documentation when you submit affidavit). Note, however, that the affidavit is submitted not to the airline – who could care less – but to US Government. And submitting a false affidavit to government is a crime. Thus you might be probed (questioning is more likely) by the CBP official on your arrival to the USA to ensure you did not submit a false affidavit.

On the other hand, the legality of the license requirement itself for private citizens to spend money overseas was never challenged in the court, and it is unclear whether our government has an authority to issue such prohibition (effectively trying to make laws for a sovereign country). The prosecution details I’ve found so far were related to hiding the fact one’s been to Cuba (i.e. lying to CBP officials), or bringing undeclared goods, such as Cuban cigars, which is a different story.

Firstly, the Cubans don’t care about the US’s restrictions at all, so you’ll have no problems with them.

When it comes to the US, anecdotal evidence suggests that US citizens won’t face legal repercussions in practice (though they may be asked what they did in Cuba and/or be yelled at)

As a foreigner, however, you are much more vulnerable – you have absolutely no rights at the US border. In essence, if they don’t like something you did, they can simply revoke your visa and status on the spot, send you home and ban you from re-entering (or even changing planes there) for many years to come (although as a visa holder, you can appeal through an immigration judge – whether that would be likely to be successful, I do not know).

Of course it will depend on the person you get at the booth, but visas have been revoked in the past for even lesser things, so the possibility, although perhaps not very high, is very real.

From the Department of State website:

The regulations require that persons subject to U.S. jurisdiction be licensed in order to engage in any travel-related transactions pursuant to travel to, from, and within Cuba, or that the transactions in question be exempt from licensing requirements. Transactions related to travel for tourist activities are not licensable. This restriction includes travel to Cuba for tourist activites from or through a third country, such as Mexico or Canada. U.S. law enforcement authorities enforce these regulations at U.S. airports and pre-clearance facilities in third countries. Travelers who fail to comply with Department of the Treasury regulations may face civil penalties and criminal prosecution upon return to the United States.

Someone I know is a US green card holder (i.e. permanent resident) and was allegedly (though it’s not yet confirmed to me) very close to being deported from the US for life after he was caught at the border with a Cuban passport stamp (in February 2016). Fortunately he’s rich and could afford an excellent solicitor, whereby the immigration judge (narrowly) decided to suspend his removal proceedings.

“Yet I do not have any official documents (from the USA or Cuba).”

You need to get something from your educational insitution. You are allowed to go to Cuba licenced, but you need to be able to prove at the US border that your visit was in fact licenced. It’s very strange that no evidence whatsoever has been provided by your insitution.